Long Memory and Fractional Integration
5.3 Fractionally Integrated Noise
108 5 Long Memory and Fractional Integration Fractional integration thus imposes the hyperbolic decay rate discussed above, where the speed of convergence varies withd. From (5.1) we observe with (5.6) thatf jgis summable if and only ifd< 0, in which case
X1 jD0
jD .1 z/ dˇˇ
zD1D0 ; d< 0 : (5.7) Further,f jgis square summable if and only ifd < 0:5. These are the ingredients to define a fractionally integrated process.
Consequently,CIRorVRdefined in (3.3) and (4.6), respectively, do not exist. Even though the MA coefficients are not summable for positived, the process is still stationary as long asd< 0:5becauseP
j 2
j <1due to (5.6) and (5.1). Further, the process is often called invertible ford> 0:5since an autoregressive representation exists that is square summable3:
X1 jD0
jxt j D"t with X1
jD0
j2 <1:
Note that the existence of an autoregressive representation in the mean square sense does not require square summability; in fact, Bondon and Palma (2007) extend the range of invertibility in the mean square to d > 1. Given the existence of the MA(1) representation the following properties of fractionally integrated noise are proven in the Problem section.
Proposition 5.1 (Fractional noise, time domain) For fractionally integrated noise from (5.8) it holds with 1 <d< 0:5that
(a) the variance equals
.0/D.0Id/D2 .1 2d/
. .1 d//2;
with ./being defined in (5.18), and.0Id/achieves its minimum for d D0;
(b) the autocovariances equal .h/D h 1Cd
h d .h 1/ ; hD1; 2; : : : ; d2h2d 1; h! 1; d¤0 ; with
d D .1 2d/
.d/ .1 d/; whered < 0if and only if d< 0;
(c) the autocorrelations.h/D.hId/grow with d for d> 0.
Let us briefly comment those results. First, since .1/ D 1, the minimum variance obtained in the white noise case (dD0) is of course.0I0/D2. Second,
3A more technical exposition can be found in Brockwell and Davis (1991, Thm. 13.2.1) or Giraitis, Koul, and Surgailis (2012, Thm. 7.2.1), although they consider only the rangejdj< 0:5.
110 5 Long Memory and Fractional Integration from the hyperbolic decay of the autocovariance sequence we observe that.h/
converges to zero withhas long asd < 0:5, but ford > 0so slowly, that we have long memory defined as
XH hD0
j .h/j! 1; H! 1 ifd> 0 : (5.9)
In particular, the autocovariances die out the more slowly the larger the memory parameterd is. Obviously, the same feature can be rephrased in terms of autocor- relations. The recursion carries over to the autocorrelations, and Proposition5.1(b) yields
.h/ d2
.0/h2d 1; h! 1:
For a numerical and graphical illustration see Fig.5.3. The asymptotic constantd
has the same sign asd, meaning that in case of long memory the autocovariances converge to zero from above, and vice versa from below zero ford < 0, see again
0 5 10 15
0 0.2 0.4 0.6 0.8 1
d=0.45
0 5 10 15
0 0.2 0.4 0.6 0.8 1
d=0.25
0 5 10 15
−0.5 0 0.5 1
d=−0.25
0 5 10 15
−0.5 0 0.5 1
d=−0.45 Fig. 5.3 .h/from Proposition5.1fordD0:45; 0:25; 0:25; 0:45
Fig.5.3. Note, however, thatdcollapses to zero asd!0, simply meaning that the hyperbolic decay rate does not hold fordD 0. Third, a similar effect thatdhas on .h/at long lags, holds true for finiteh. More precisely, Proposition5.1(c) says for each finitehthat the autocorrelation grows withd(ford > 0), which reinforces the interpretation ofdas measure of persistence or long memory.4
The case of negativedresults in short memory in that the autocovariances are absolutely summable, which is clear again from thep-series in (5.1). This case is sometimes called antipersistent, the reason for that being
X1 jD1
jD0 ; ifd< 0 :
This property translates into a special case of short memory, namely X1
hD 1
.h/D0 ; ifd< 0 ;
as we will become obvious from the spectrum at frequency zero.
Long Memory in the Frequency Domain
It is obvious from the definition of the spectrum in (4.3) that it does not exist at the origin under long memory (d > 0), because the autocovariances are not summable.
Still, the previous chapter has been set up sufficiently general to cover long memory, see (4.1). Given a singularity at frequencyD 0, one still may determine the rate at whichf./goes off to infinity asapproaches 0. To determinef, we have to evaluate the power transfer function of.1 L/ dfrom Proposition4.1and obtain5
T.1 L/ d./D.2 2cos.// dD
4sin2
2 d
;
4More complicated is the effect of changes indifd< 0, see Hassler (2014).
5Readers not familiar with complex numbers,i2D 1, may skip the following equation, see also Footnote6in Chap.4:
T.1 L/ d./D.1 ei/ d.1 e i/ d D.1 ei e iC1/ d D.2 2cos.// d:
112 5 Long Memory and Fractional Integration
where the trigonometric half-angle formula was used for the second equality:
2sin2.x/D1 cos.2x/ : (5.10)
We hence have the following result.
Proposition 5.2 (Fractional noise, frequency domain) Under the assumptions of Proposition5.1it holds for the spectrum of fractional noise xt D.1 L/ d"tthat
f./D
4sin2
2 d
2
2; > 0 ; (5.11)
and
f./ 2d2
2; !0 : (5.12)
The second statement in Proposition5.2is again understood to be asymptotic:
Similarly to (5.2) we denote for two functiona.x/andb.x/¤0:
a.x/b.x/forx!0 ” lim
x!0
a.x/
b.x/ D1 : (5.13)
Since limx!0sin.x/=x D 1 we write sin.x/ xforx !0. Consequently, (5.12) arises from (5.11).
From Proposition5.2we learn that long memory (d> 0) translates into a spectral singularity at frequency zero, and the negative slope is the steeper the largerd is. In other words: the longer the memory, the stronger is the contribution of the long-run trend to the variance of the process. The antipersistent case in contrast is characterized by the opposite extreme:f.0/D0. For an illustration, have a look at Fig.5.4.
Example 5.1 (Fractionally Integrated Noise) Although the fractional noise is dom- inated by the trend component at frequency zero (strongly persistent) ford > 0, the process is stationary as long asd < 0:5. Consequently, a typical trajectory can not drift off but displays somehow reversing trends. In Fig.5.5 we see from simulated data that the deviations from the zero line are stronger ford D 0:45 than ford D 0:25. The antipersistent series (d D 0:45), in contrast, displays an oscillating behavior due to the negative autocorrelation.
0 1 2 3 0
2 4 6 8
d=0.45
0 1 2 3
0 2 4 6 8
d=0.25
0 1 2 3
0 2 4 6 8
d=−0.25
0 1 2 3
0 2 4 6 8
d=−0.45 Fig. 5.4 2f./from Proposition5.2fordD0:45; 0:25; 0:25; 0:45